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Metamath Proof Explorer

Theorem cnmpt2ds

Description: Continuity of the metric function; analogue of cnmpt22f which cannot be used directly because D is not necessarily a function. (Contributed by Mario Carneiro, 5-Oct-2015)

Ref Expression
Hypotheses cnmpt1ds.d ⊢ 𝐷 = ( dist ‘ 𝐺 )
cnmpt1ds.j ⊢ 𝐽 = ( TopOpen ‘ 𝐺 )
cnmpt1ds.r ⊢ 𝑅 = ( topGen ‘ ran (,) )
cnmpt1ds.g ⊢ ( 𝜑 → 𝐺 ∈ MetSp )
cnmpt1ds.k ⊢ ( 𝜑 → 𝐾 ∈ ( TopOn ‘ 𝑋 ) )
cnmpt2ds.l ⊢ ( 𝜑 → 𝐿 ∈ ( TopOn ‘ 𝑌 ) )
cnmpt2ds.a ⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) ∈ ( ( 𝐾 ×t 𝐿 ) Cn 𝐽 ) )
cnmpt2ds.b ⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐵 ) ∈ ( ( 𝐾 ×t 𝐿 ) Cn 𝐽 ) )
Assertion cnmpt2ds ( 𝜑 → ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ ( 𝐴 𝐷 𝐵 ) ) ∈ ( ( 𝐾 ×t 𝐿 ) Cn 𝑅 ) )

Proof

Step Hyp Ref Expression
1 cnmpt1ds.d ⊢ 𝐷 = ( dist ‘ 𝐺 )
2 cnmpt1ds.j ⊢ 𝐽 = ( TopOpen ‘ 𝐺 )
3 cnmpt1ds.r ⊢ 𝑅 = ( topGen ‘ ran (,) )
4 cnmpt1ds.g ⊢ ( 𝜑 → 𝐺 ∈ MetSp )
5 cnmpt1ds.k ⊢ ( 𝜑 → 𝐾 ∈ ( TopOn ‘ 𝑋 ) )
6 cnmpt2ds.l ⊢ ( 𝜑 → 𝐿 ∈ ( TopOn ‘ 𝑌 ) )
7 cnmpt2ds.a ⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) ∈ ( ( 𝐾 ×t 𝐿 ) Cn 𝐽 ) )
8 cnmpt2ds.b ⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐵 ) ∈ ( ( 𝐾 ×t 𝐿 ) Cn 𝐽 ) )
9 txtopon ⊢ ( ( 𝐾 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐿 ∈ ( TopOn ‘ 𝑌 ) ) → ( 𝐾 ×t 𝐿 ) ∈ ( TopOn ‘ ( 𝑋 × 𝑌 ) ) )
10 5 6 9 syl2anc ⊢ ( 𝜑 → ( 𝐾 ×t 𝐿 ) ∈ ( TopOn ‘ ( 𝑋 × 𝑌 ) ) )
11 mstps ⊢ ( 𝐺 ∈ MetSp → 𝐺 ∈ TopSp )
12 4 11 syl ⊢ ( 𝜑 → 𝐺 ∈ TopSp )
13 eqid ⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 )
14 13 2 istps ⊢ ( 𝐺 ∈ TopSp ↔ 𝐽 ∈ ( TopOn ‘ ( Base ‘ 𝐺 ) ) )
15 12 14 sylib ⊢ ( 𝜑 → 𝐽 ∈ ( TopOn ‘ ( Base ‘ 𝐺 ) ) )
16 cnf2 ⊢ ( ( ( 𝐾 ×t 𝐿 ) ∈ ( TopOn ‘ ( 𝑋 × 𝑌 ) ) ∧ 𝐽 ∈ ( TopOn ‘ ( Base ‘ 𝐺 ) ) ∧ ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) ∈ ( ( 𝐾 ×t 𝐿 ) Cn 𝐽 ) ) → ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) : ( 𝑋 × 𝑌 ) ⟶ ( Base ‘ 𝐺 ) )
17 10 15 7 16 syl3anc ⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) : ( 𝑋 × 𝑌 ) ⟶ ( Base ‘ 𝐺 ) )
18 eqid ⊢ ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) = ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 )
19 18 fmpo ⊢ ( ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑌 𝐴 ∈ ( Base ‘ 𝐺 ) ↔ ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) : ( 𝑋 × 𝑌 ) ⟶ ( Base ‘ 𝐺 ) )
20 17 19 sylibr ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑌 𝐴 ∈ ( Base ‘ 𝐺 ) )
21 20 r19.21bi ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → ∀ 𝑦 ∈ 𝑌 𝐴 ∈ ( Base ‘ 𝐺 ) )
22 21 r19.21bi ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑦 ∈ 𝑌 ) → 𝐴 ∈ ( Base ‘ 𝐺 ) )
23 cnf2 ⊢ ( ( ( 𝐾 ×t 𝐿 ) ∈ ( TopOn ‘ ( 𝑋 × 𝑌 ) ) ∧ 𝐽 ∈ ( TopOn ‘ ( Base ‘ 𝐺 ) ) ∧ ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐵 ) ∈ ( ( 𝐾 ×t 𝐿 ) Cn 𝐽 ) ) → ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐵 ) : ( 𝑋 × 𝑌 ) ⟶ ( Base ‘ 𝐺 ) )
24 10 15 8 23 syl3anc ⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐵 ) : ( 𝑋 × 𝑌 ) ⟶ ( Base ‘ 𝐺 ) )
25 eqid ⊢ ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐵 ) = ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐵 )
26 25 fmpo ⊢ ( ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑌 𝐵 ∈ ( Base ‘ 𝐺 ) ↔ ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐵 ) : ( 𝑋 × 𝑌 ) ⟶ ( Base ‘ 𝐺 ) )
27 24 26 sylibr ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑌 𝐵 ∈ ( Base ‘ 𝐺 ) )
28 27 r19.21bi ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → ∀ 𝑦 ∈ 𝑌 𝐵 ∈ ( Base ‘ 𝐺 ) )
29 28 r19.21bi ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑦 ∈ 𝑌 ) → 𝐵 ∈ ( Base ‘ 𝐺 ) )
30 22 29 ovresd ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑦 ∈ 𝑌 ) → ( 𝐴 ( 𝐷 ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝐵 ) = ( 𝐴 𝐷 𝐵 ) )
31 30 3impa ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ) → ( 𝐴 ( 𝐷 ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝐵 ) = ( 𝐴 𝐷 𝐵 ) )
32 31 mpoeq3dva ⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ ( 𝐴 ( 𝐷 ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝐵 ) ) = ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ ( 𝐴 𝐷 𝐵 ) ) )
33 13 1 2 3 msdcn ⊢ ( 𝐺 ∈ MetSp → ( 𝐷 ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ∈ ( ( 𝐽 ×t 𝐽 ) Cn 𝑅 ) )
34 4 33 syl ⊢ ( 𝜑 → ( 𝐷 ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ∈ ( ( 𝐽 ×t 𝐽 ) Cn 𝑅 ) )
35 5 6 7 8 34 cnmpt22f ⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ ( 𝐴 ( 𝐷 ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝐵 ) ) ∈ ( ( 𝐾 ×t 𝐿 ) Cn 𝑅 ) )
36 32 35 eqeltrrd ⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ ( 𝐴 𝐷 𝐵 ) ) ∈ ( ( 𝐾 ×t 𝐿 ) Cn 𝑅 ) )